Introduction | |

Mathematical Models, Solutions, and Direction Fields | |

Linear Equations: Method of Integrating Factors | |

Numerical Approximations: Euler's Method | |

Classification of Differential Equations | |

First Order Differential Equations | |

Separable Equations | |

Modeling with First Order Equations | |

Differences between Linear and Nonlinear Equations | |

Autonomous Equations and Population Dynamics | |

Exact Equations and Integrating Factors | |

Accuracy of Numerical Methods | |

Improved Euler and Runge-Kutta Methods | |

Projects | |

Harvesting a Renewable Resource | |

Designing a Drip Dispenser for a Hydrology Experiment | |

A Mathematical Model of a Groundwater Contaminant Source | |

Monte-Carlo Option Pricing: Pricing Financial Options by Flipping a Coin | |

Systems of Two First Order Equations | |

Systems of Two Linear Algebraic Equations | |

Systems of Two First Order Linear Differential Equations | |

Homogeneous Linear Systems with Constant Coefficients | |

Complex Eigenvalues | |

Repeated Eigenvalues | |

A Brief Introduction to Nonlinear Systems | |

Numerical Methods for Systems of First Order Equations | |

Projects | |

Eigenvalue Placement Design of a Satellite Attitude Control System | |

Estimating Rate Constants for an Open Two-Compartment Model | |

The Ray Theory of Wave Propagation | |

A Blood-Brain Pharmacokinetic Model | |

Second Order Linear Equations | |

Definitions and Examples | |

Theory of Second Order Linear Homogeneous Equations | |

Linear Homogeneous Equations with Constant Coefficients | |

Mechanical and Electrical Vibrations | |

Nonhomogeneous Equations: Method of Undetermined Coefficients | |

Forced Vibrations, Frequency Response, and Resonance | |

Variation of Parameters | |

Projects | |

A Vibration Insulation Problem | |

Linearization of a Nonlinear Mechanical System | |

A Spring-Mass Event Problem | |

Uniformly Distributing Points on a Sphere | |

Euler-Lagrange Equations | |

The Laplace Transform | |

Definition of the Laplace Transform | |

Properties of the Laplace Transform | |

The Inverse Laplace Transform | |

Solving Differential Equations with Laplace Transforms | |

Discontinuous Functions and Periodic Functions | |

Differential Equations with Discontinuous Forcing Functions | |

Impulse Functions | |

Convolution Integrals and Their Applications | |

Linear Systems and Feedback Control | |

Projects | |

An Electric Circuit Problem | |

Effects of Pole Locations on Step Responses of Second Order Systems | |

The Watt Governor, Feedback Control, and Stability | |

Systems of First Order Linear Equations | |

Definitions and Examples | |

Basic Theory of First Order Linear Systems | |

Homogeneous Linear Systems with Constant Coefficients | |

Complex Eigenvalues | |

Fundamental Matrices and the Exponential of a Matrix | |

Nonhomogeneous Linear Systems | |

Defective Matrices | |

Projects | |

A Compartment Model of Heat Flow in a Rod | |

Earthquakes and Tall Buildings | |

Controlling a Spring-Mass System to Equilibrium | |

Nonlinear Differential Equations and Stability | |

Almost Linear Systems | |

Competing Species | |

Predator-Prey Equations | |

Periodic Solutions and Limit Cycles | |

Chaos and Strange Attractors: The Lorenz Equations | |

Projects | |

Modeling of Epidemics | |

Harvesting in a Competitive Environment | |

The Rossler System | |

[Chapters 8-10 in Boundary Value Problems version only] | |

Series Solutions of Second Order Equations | |

Review of Power Systems | |

Series Solutions Near an Ordinary Point, Part I | |

Series Solutions Near an Ordinary Point, Part II | |

Regular Singular Points | |

Series Solutions Near a Regular Singular Point, Part I | |

Series Solutions Near a Regular Singular Point, Part II | |

Bessel's Equation | |

Projects | |

Distraction Through a Circular Aperture | |

Hermite Polynomials and the Quantum Mechanical Harmonic Oscillator | |

Perturbation Methods | |

Partial Differential Equations and Fourier Series | |

Two-Point Boundary Value Problems | |

Fourier Series | |

The Fourier Convergence Theorem | |

Even and Odd Functions | |

Separation of Variables, Heat Conduction in a Rod | |

Other Heat Conduction Problems | |

The Wave Equation, Vibrations of an Elastic String | |

Laplace's Equation | |

Projects | |

Estimating the Diffusion Coefficient in the Heat Equation | |

The Transmission Line Problem | |

Solving Poisson's Equation by Finite Differences | |

Matrices and Linear Algebra | |

Matrices | |

Systems of Linear Algebraic Equations, Linear Independence, and Rank | |

Determinants and Inverses | |

The Eigenvalue Problem | |

Complex Variables | |

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